Best possible upper bounds on the restrained domination number of cubic graphs

A dominating set in a graph $G$ is a set $S$ of vertices such that every vertex in $V\left(G\right)⧹S$ is adjacent to a vertex in $S$. A restrained dominating set of $G$ is a dominating set $S$ with the additional restraint that the graph $G-S$ obtained by removing all vertices in $S$ is isolate-free. The domination number $\gamma \left(G\right)$ and the restrained domination number ${\gamma }_r\left(G\right)$ are the minimum cardinalities of a dominating set and restrained dominating set, respectively, of $G$. Let $G$ be a cubic graph of order $n$. A classical result of Reed states that $\gamma \left(G\right)\le \frac{3}{8}n$, and this bound is best possible. To determine the best possible upper bound on the restrained domination number of $G$ is more challenging, and we prove that ${\gamma }_r\left(G\right)\le \frac{2}{5}n$.